Start with the unit circlev2+w2=1
centred at the origin and with radius one. We can give a closed formula
for a point (v,w) on the circle. A typical point is(v,w)=(1-t2) / (1+t2), 2t / (1+t2)) (*).

The proof of this assertion is a consequence of those trigonometric
identities I mentioned but a more geometric point is given as follows
(you will want to draw a diagram to understand this).
We project the circle onto the w-axis. Consider the line that
goes through the point (-1,0) and hits the w-axis at the
point (0,t). Where does this line meet the circle? Well it meets
it exactly at the point (*) above.

It's clear that this point (v,w) is a pair of rational numbers
if and only ift is rational. So to get back to the proof of the theorem,
suppose that we have a primitive Pythagorean triple (a,b,c). Then
(a/c,b/c) is a point on the unit circle. So by the above argument
there corresponds a rational number t to this point. Write
t=x/y in its lowest terms (so that x and y are
coprime). The formula (*) nows gives us that
a=x2-y2, b=2xy and c=x2+y2.
This shows that every primitive triple has the required form. That
triples of the form in the statement are primitive Pythagorean triples
is clear, so we are done.